<h2>Problem 285</h2>
<div style="color:#666;font-size:80%;">03 April 2010</div><br />
<div class="problem_content">
<p>Albert chooses a positive integer <var>k</var>, then two real numbers <var>a</var>, <var>b</var> are randomly chosen in the interval [0,1] with uniform distribution.<br />
The square root of the sum (<var>k</var>&middot;<var>a</var>+1)<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" />&thinsp;+&thinsp;(<var>k</var>&middot;<var>b</var>+1)<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> is then computed and rounded to the nearest integer. If the result is equal to <var>k</var>, he scores <var>k</var> points; otherwise he scores nothing.</p>

<p>For example, if <var>k</var>&thinsp;=&thinsp;6, <var>a</var>&thinsp;=&thinsp;0.2 and <var>b</var>&thinsp;=&thinsp;0.85, then (<var>k</var>&middot;<var>a</var>+1)<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" />&thinsp;+&thinsp;(<var>k</var>&middot;<var>b</var>+1)<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" />&thinsp;=&thinsp;42.05.<br />
The square root of 42.05 is 6.484... and when rounded to the nearest integer, it becomes 6.<br />
This is equal to <var>k</var>, so he scores 6 points.</p>

<p>It can be shown that if he plays 10 turns with <var>k</var>&thinsp;=&thinsp;1, <var>k</var>&thinsp;=&thinsp;2, ..., <var>k</var>&thinsp;=&thinsp;10, the expected value of his total score, rounded to five decimal places, is 10.20914.</p>

<p>If he plays 10<img src="" style="display:none;" alt="^(" /><sup>5</sup><img src="" style="display:none;" alt=")" /> turns with <var>k</var>&thinsp;=&thinsp;1, <var>k</var>&thinsp;=&thinsp;2, <var>k</var>&thinsp;=&thinsp;3, ..., <var>k</var>&thinsp;=&thinsp;10<img src="" style="display:none;" alt="^(" /><sup>5</sup><img src="" style="display:none;" alt=")" />, what is the expected value of his total score, rounded to five decimal places?</p>
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